Transformations
by ntepAC
Summary: Ezio  unwillingly  takes a stab at something a killer normally doesn't do... Math. And who else but Leonardo to teach him some new math? Rated K  for language


_A/N: Ezio Auditore da Firenze and Assassin's Creed (I and II) belongs to Ubisoft and Ubisoft Montreal, and Leonardo da Vinci belongs to himself. :D_

_(Alright, so I'm not dead, as though I'm sure some in the world would wish. ^^" I kid. Anywho, so this was a project I had to do in PreCalculus. I had to come up with some sort of way to explain graphical transformations to someone else. And I thought, why not the great Leonardo explain it to the assassin Ezio? LOL, so here's the fruit of it. I got 25/25 on it, so I'm proud of it! ^^ And there are some minor errors, like sporadic punctuation, but that was cause I was in a rush, and I'll fix it when I can... So please overlook it! ^^")_

_Assassin's Creed II takes place during renaissance Italy, so NORMALLY they don't have graphing paper like we do today. But I'm bending it a TAD bit to fit the requirements. : )_

**Transformations**

"Va bene… and you go here."

The assassin roiled his brows. "Uh… What are you… Never mind. I've another codex page for you."

"The artist/inventor/mathematician smiled. "how exciting! Yet another one, Ezio!" He stood up and took the page from Ezio.

"Yep." Ezio nodded. "But, what were you doing, anyway, Leonardo?"

"Transformations." The artist smiled.

The assassin's brows roiled again. "Transformations?"

"Si. When you change a graph using a function."

Ezio's face flashed blank as he lifted his hands to his head, his index fingers near his ears. "In," he twitched his right index finger, "out." Then his left.

Leonardo laughed. "Oh, Ezio, amico mio. Shall I teach you about transformations?"

The assassin's hands returned to his sides as he seemed to wince. "If I say no, you're going to teach me anyway."

"Ma certo!" Leonardo smiled innocently.

Ezio nodded. "Va bene. Teach me."

"Bene! Bene!" The artist sat down and handed Ezio a pencil and piece of graph paper. "So, here's your parent graph of…"

"Woah, wait. What's a parent graph?" Ezio asked.

"Mi dispiace, I'm getting ahead of myself. A parent graph is the graph of one of the twelve basic functions."

"Twelve?"

"Si. So your parent graph will be… Hmm… The Sine function." Leonardo thought for a minute, plotted the points, and drew the graph for Ezio. "Did you know that this function and your sinus cavities," the artist tapped the bridge of his nose, "were named from the common Latin root word for "bay"? I believe Robert of Chester made this mistake mistranslating it…"

Ezio stared blankly at Leonardo.

"I'm sorry…" Leonardo chuckled. "It's interesting to me. So your formula is…" Leonardo wrote down what looked like gibberish to Ezio. "Let f of x equal a, f, b, x minus c, plus d."

_f(x) = a(f(b(x-c))) + d_

Ezio roiled his brows again. "What?"

Leonardo smiled. "That is the formula you use when changing a graph using a function. So we're going to change the function like this." He wrote more gibberish. "Negative one-half x plus five minus two."

_f(x) = -1/2(x-5)+2_

Ezio stared at the paper.

Leonardo stared at Ezio. "Perhaps I need to start a bit easier for you…"

"Si…" Ezio nodded slowly.

"No problem!" Leonardo crumpled up the paper and tossed it aside. "Alright, so let's use the squaring function."

_f(x) = x__2 _

He grabbed a new piece of paper and set up the new grid. "To create the squaring function you set up a table."

A wooden table popped into Ezio's head. "Huh?"

"A number table, amico." Leonardo drew a large 't' and put 'x' in the upper left-hand corner and 'x-squared' in the upper right hand corner.

"Oh!"

"Yes!" Leonardo smiled, which quickly vanished. "What were you…"

"Nothing, continue."

The artist shrugged. "Alright. So, in the 'x' column you write three, two, one, zero, negative one, negative two, and negative three."

Ezio obeyed, writing the numbers as Leonardo said them.

"Bene. And now you square each of those numbers. Like three squared would be?"

"Nine." Ezio wrote nine under the 'x-squared' column.

"Good. I assume you can square the rest?"

"Si." Ezio nodded and squared the rest of the numbers.

"Bene. Now, you have your 'x's' and your 'y's'. So plot the points."

"I can do basic math in my head Leonardo… I know nothing about graphs."

"Alright. So this," Leonardo pointed to the bold horizontal line, "is the x-axis, and this, "He pointed to the bold vertical line, "is the y-axis."

"Oh. And the numbers I got go on her somehow?"

"Yep." Leonardo nodded.

"So which one's the y's?"

"The x-squared's." Leonardo pointed to the x-squared column.

"okay…"

Leonardo showed Ezio how to plot the points, eventually getting a 'u' shaped line on the graphe. "Va bene. So this is called a parabola."

Ezio repeated the word. "Parabola. Okay."

"So let's return to that formula I showed you." Leonardo wrote the formula again. "Okay. Let's start with 'c'. It moves the graph horizontally. So if I were to write x minus two squared."

_f(x) = (x-2)__2_

"Where'd the other stuff go?"

"Because we only have a value for 'c', the other stuff is zero. So we don't worry about those right now."

Ezio nodded. "Alright."

"So, everything in the parenthesis is not what it seems… it's the opposite."

"So that means negative two is positive two?"

"Bingo!" Leonardo smiled.

"Okay… so…"

"We move the parabola two units to the right."

Ezio looked at each point he had plotted and moved them to the right two units. He redrew the parabola.

"Good. You got it."

Ezio nodded. "Alright."

Leonardo grabbed a different color of pencil and handed it to Ezio. "Now, we'll do something different to the parabola. Let's say I write x-squared minus three…"

_f(x) = x__2__-3_

"What value do we have now?"

"Not 'c', right?"

"Si."

"Ezio stared at the formula.

"The three isn't grouped with the x… it's at the end."

"Oh, 'd'."

Leonardo smiled. "Good. And what is 'd'?"

"Three?"

"It's not in the parenthesis. So it's what you see."

"So, negative three?"

"Good. Now 'd' moves the graph vertically."

"So we move it down three units?"

"Yes! You're getting it!"

"Hmm… I guess I am…" Ezio smiled slightly.

"So, since you understand 'c' and 'd', le'ts work with 'a'." Leonardo wrote another function. "So this one affects the graph vertically also. But, instead of moving it, it stretches it, or shrinks it, depending on its value. So, two x-squared."

_2f(x) = 2x__2_

"Ready Ezio? This one's a bit tricky."

Ezio nodded. "Si."

"Va bene. So, if 'a' is bigger than one, it's a vertical stretch. If 'a' is smaller than one, it's a vertical shrink."

"So we have a stretch here, right?"

"Yes, yes!" Leonardo smiled.

"So, what do we do?"

"We multiply the y's by a factor of two."

"Why the y's?" Ezio stared at his graph.

"Vertical, Ezio. The y-axis is vertical."

"Oh…" Ezio blushed a bit.

"Quite alright. You're getting quite good at this, so don't put yourself down." Leonardo smiled encouragingly. "So, we take our original table, and we multiply each 'y' by two."

Ezio scribbled on the paper the new points, then plotted each point, drawing the new parabola. "It got skinnier."

"Si. It was _stretched_, like a rubber band." Leonardo pulled out a rubber band and stretched it, aiming at Ezio.

Ezio cowered, "Hey! Hey! Don't shoot!"

Leonardo laughed. "I'm kidding! So, let's change the two to one-half."

_1/2f(x) = 1/2x__2_

"Alright. One-half is smaller than one, so it's a shrink?"

"Si. A vertical shrink."

"So, do I multiply each 'y' by one-half?"

"Yes, or divide by two."

Ezio nodded and scribbled on the paper again, creating new points, and plotting new points. "It seems squished."

Leonardo smiled and nodded.

"So, let's go to 'b'." Ezio smiled. "I've got the hang of this."

Leonardo stared at the paper. "Merda! I forgot one thing!"

"What is that?"

"Let's put a negative sign in front of the x."

_-f(x) = -x__2_

"What's that do?"

"We reflect the parabola over the x-axis. So we invert each 'y' by multiplying each value by negative one. Our nine would become negative nine, four to negative four, so on and so forth."

"Oh…" Ezio nodded. "Okay." He scribbled, plotted points, and drew the flipped parabola.

"Good." The artist smiled. "I knew you'd understand that. So let's move on to 'b'." His left hand scribbled furiously on the paper a new function. "Two x-squared."

_f(2x) = (2x)__2_

"It's the same." Ezio frowned, roiling his brows.

"It's not the same." Leonardo smiled. "Last time, technically the two wasn't groupled with the x. This time, it is."

Ezio stared blankly at Leonardo. "And I thought I had this…"

"I told you 'a' and 'b' were tricky."

"You told me 'a' was tricky!" Ezio growled.

"Alright, 'b' is even more tricky."

Ezio growled. "I think Il Magnifico might have something new for me…"

"No, no, no, Ezio! You're almost done. Just a little more, per favore?" The artist pleaded.

Ezio sighed. "Va bene."

"Grazie!" Leonardo smiled. "So, 'b' stretches and shrinks the graph horizontally. Also, what ever the value for 'b' is, its effect is the reciprocal. So that means we flip the number."

"So it's a negative?"

"No, no. We put it under one."

"Oh…" Ezio roiled his brows.

"Alright, so the two is technically over one right now, two one over one's equal two."

"uh-huh." Ezio nodded.

"So if we flip two over one, we get…"

"One-half!" Ezio's face lit up. "Okay."

"Great! So we divide each 'x' by two."

"Oh, cause it's horizontal."

"Yes! See? You've got it!"

"Alright…" Ezio scribbled and plotted. "It got skinnier…"

"It shrunk. You shrunk the graph by a factor of one-half, because two is bigger than one."

"Okay… so if we have one-half x squared…"

_ f(1/2x) = (1/2x)__2_

Leonardo smiled widely, stepping back and watching Ezio work.

"We'll multiply the 'x' by the reciprocal, which will be two…" The assassin scribbled and plotted. "And I stretched it, because one-half is smaller than one." Ezio smiled. He looked at Leonardo. "So if I applied a negative…"

"It would flip over the y-axis." Leonardo flipped his hands over the invisible y-axis he drew in the air.

"Okay." Ezio smiled. "I got this!"

"Ready to apply it all?"

Ezio nodded.

"Great!" Leonardo grabbed a new piece of graph paper and drew the grid, plotted some points, made what seemed to be a random graph, and put it down over the old graph paper with the old, worn out parabola. "Here's your graph. Your points are three-two, zero-one, negative two-zero, and negative four-negative three."

_(3,2)_

_ (0,1)_

_ (-2,0)_

_ (-4,-3)_

"And your function is…" Leonardo thought. "Ah! Negative 'f', two, x plus one, plus two…"

_-f(2(x+1))+2_

"So, first, what are your values?"

Ezio stared at the new function, writing on the new paper. "Um… 'a' is negative one, 'b' is two, 'c' is… negative one?"

"Good. A negative minus a number is really adding those numbers together." The artist nodded.

"And 'd' is two."

"Good. Now we map the points. Let's apply the negative first. Is 'a' vertical or horizontal?"

"Vertical."

"Yes. So we change?"

"They y's."

"Good."

Ezio wrote the opposites of each y-value.

_(3,2) - (3,-2)_

_ (0,1) - (0,-1)_

_ (-2,0) - (-2,0)_

_ (-4,-3) - (-4,3)_

"Now let's apply 'b'. What's 'b'?"

"Two, so we divide by two."

"Good. Do we flip the graph?"

"No, there is no negative."

"Good."

Ezio divided each 'x' by two.

_ (3,2) - (3,-2) - (1.5,-2)_

_ (0,1) - (0,-1) - (0,-1)_

_ (-2,0) - (-2,0) - (-1,0)_

_ (-4,-3) - (-4,3) - (-2, 3)_

"Good. Apply 'c'. What happens?"

"We subtract one from each 'x'."

Leonardo nodded as Ezio subtracted.

_ (3,2) - (3,-2) - (1.5,-2) - (.5,-2)_

_ (0,1) - (0,-1) - (0,-1) - (-1,-1)_

_ (-2,0) - (-2,0) - (-1,0) - (-2,0)_

_ (-4,-3) - (-4,3) - (-2, 3) - (-3,3)_

"Apply 'd'." Leonardo looked expectantly at Ezio.

"We add two to each 'y'."

The artist nodded.

_ (3,2) - (3,-2) - (1.5,-2) - (.5,-2) - (.5,0)_

_ (0,1) - (0,-1) - (0,-1) - (-1,-1) - (-1,1)_

_ (-2,0) - (-2,0) - (-1,0) - (-2,0) - (-2,2)_

_ (-4,-3) - (-4,3) - (-2, 3) - (-3,3) - (-3,5)_

"Now, plot your new points and draw your new graph."

Ezio nodded and plotted.

"What happened to it?"

Ezio looked at the new graph. "We flipped it over the x-axis, shrank it horizontally, moved it to the left one unit and up two units."

"Bene! You got it!" Leonardo smiled.

Ezio nodded, his interest in the subject now gone. "How about that Codex page?" The assassin smiled innocently.

_Italian Vocab:_

_Va bene – Very well/alright. Merda – Sh*t_

_ Amico mio – My friend_

_ Ma certo! – But of course!_

_ Bene – good_

_ Mi dispiace – My apologies/I'm sorry._

_ Si – yes_


End file.
